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Well-Ordering Principle : Every non-empty set of positive integers contains a least element

Least Upper Bound Property: Every nonempty bounded subset of the real numbers has a least upper bound.

Is the Well-Ordering derived from the LUB property?

Lemon
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1 Answers1

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I would not derive a basic property of the natural numbers, like the well-ordering principle, from a more complicated property of the more complicated system of real numbers, like the least upper bound principle. I believe that most if not all systematic presentations of the relevant foundations agree with me and prove well-ordering of the natural numbers before doing anything with the real numbers.

To also answer your title question: The difference is that (1) they are about different systems --- the natural numbers in one case and the real numbers in the other --- and (2) they assert very different things about these systems --- existence of a smallest element of a set in the one case and existence of an element "near" the top of a set in the other.

Perhaps a better question would be what are the similarities between these two principles. I can think of just two: First, any well-ordering has the least-upper-bound property, even without the "nonempty" hypothesis (but note that this does not help in proving the least-upper-bound property for the real numbers, as these are not well-ordered). Second, both properties involve a quantifier over arbitrary subsets of the system, so they cannot be expressed fully in first-order logic but need second-order logic.

Andreas Blass
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  • But I can actually get the result of the Well-Ordering from LUB right? Because every subset of the positive integers (they must be bounded) has a least upper bound or glb. – Lemon Feb 23 '14 at 06:36
  • @sidht: If you insist, then yes. If $A$ is a set of positive integers then we can consider $S={r\in\Bbb R\mid\forall a\in A:\lfloor r\rfloor<a}$, and consider its least upper bound. But as Andreas stresses out, these are two different systems. And I agree. Mistaking the natural numbers to be a subsystem of the real numbers is a first step into assuming (mistakenly) that the collection of sets of integers, and by extension the real numbers, are countable. (The root of that mistake is that it seems that somehow continuity needs to be involved there.) – Asaf Karagila Feb 23 '14 at 06:53
  • I am guessing his worry is that $\sup $ may not belong to the set? But in my example (first comment), there is no fear, because my set is going to finite (and hence closed). – Lemon Feb 23 '14 at 07:26